1. Updated status of the PSB impedance model
C. Zannini and G. Rumolo
Thanks to: E. Benedetto, N. Biancacci, E. Métral,
B. Mikulec, N. Mounet, T. Rijoff, B. Salvant, W. Weterings

2. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

3. Impedance calculations for b < 1
Analytical calculation (applies only to simple structures)
3D EM simulation (CST Particle Studio)
In the LHC, SPS, PS CST EM simulation are performed in the ultra-relativistic approximation (b = 1)
The use of 3D EM simulations for b < 1 is not straightforward at all

4. 3D CST EM simulation for b < 1
Depend only on the source
contribution due to the interaction of beam and external surroundings
To single out the impedance contribution the direct space charge must be removed

5. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

6. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Beam pipe
Specificare che la cavita’ e’ quella a 200 MHz

7. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Beam pipe
Kickers
Specificare che la cavita’ e’ quella a 200 MHz

8. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Indirect space charge impedance
Beam pipe
Kickers
Specificare che la cavita’ e’ quella a 200 MHz
Indirect space charge impedance

9. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Indirect space charge impedance
Beam pipe
Kickers
Specificare che la cavita’ e’ quella a 200 MHz
Indirect space charge impedance

10. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Indirect space charge impedance
Beam pipe
Kickers
Specificare che la cavita’ e’ quella a 200 MHz
Indirect space charge impedance

11. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Indirect space charge impedance
Beam pipe
Kickers
Specificare che la cavita’ e’ quella a 200 MHz
Indirect space charge impedance

12. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Indirect space charge impedance
Broadband impedance of step transitions
Beam pipe
Kickers
Specificare che la cavita’ e’ quella a 200 MHz
Indirect space charge impedance
Step transitions

13. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Indirect space charge impedance
Broadband impedance of step transitions
KSW magnets
Beam pipe
Extraction Kicker
Indirect space charge impedance
Step transitions
KSW

14. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

15. Indirect space charge
Analytical calculation based on the PSB aperture model (provided by O. Berrig)
[a, b, L, βx , βy , Apertype]i
Rectangular pipe
Circular pipe
K. Y. Ng, Space charge impedances of beams with non-uniform transverse distributions
Elliptic pipe

16. PSB: indirect space charge impedance

17. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

18. Indirect space charge: refinement of the calculation
Using numerical form factors

19. Indirect space charge: refinement of the calculation
Using numerical form factors

20. PSB: indirect space charge impedance

21. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

22. Resistive wall impedance
Wall thickness
Wall (σel)
Background material
Dipoles
0.4 mm
S/m
Iron (silicon steel)
Quadrupoles
1.5 mm
S/m
Straight sections
1 mm
Vacuum
Analytical calculation based on the PSB aperture model (provided by O. Berrig)
[a, b, L, βx , βy , Apertype]i
Calculation performed with the TLwall code

23. Resistive wall impedance
Wall thickness
Wall (σel)
Background material
Dipoles
0.4 mm
S/m
Iron (silicon steel)
Quadrupoles
1.5 mm
S/m
Straight sections
1 mm
Vacuum
Analytical calculation based on the PSB aperture model (provided by O. Berrig)
Vertical
[a, b, L, βx , βy , Apertype]i
Calculation performed with the TLwall code

24. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

25. A theoretical calculation for the C-Magnet model
is the impedance calculated using the Tsutsui formalism
Constant horizontal impedance

26. PSB extraction kicker: impedance due to the ferrite loaded structure ZM
Vertical

27. PSB extraction kicker: impedance due to the coupling to the external circuits ZTEM
Horizontal
Cables in the open-short configuration

28. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

29. Broadband impedance of a step transition
Based on the results of 3D EM simulations, the broadband impedance contribution due to an abrupt transition is independent of the relativistic beta. Therefore, based on the aperture model, the generalized broadband impedance of the PSB transitions has been calculated as:
C. Zannini, Electromagnetic simulations of CERN accelerator components and experimental applications. PhD thesis, Lausanne, EPFL, CERN-THESIS

30. Broadband impedance of step transitionsL
Weak dependence on the relativistic beta and L

31. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

32. 3D model of the KSW kicker

33. Coating of the ceramic chamber

34. Coating of the ceramic chamber

35. Analytical model: axially symmetric multilayer structure
Assuming uniformity of the coating thickness
Resistance of the coating
The beam coupling impedance strongly depends on the resistance of the coating

36. Effect of the ceramic chamber on the KSW longitudinal impedance
Titanium coating
The shielding (coated ceramic chamber) strongly reduces the longitudinal impedance

37. Effect of the ceramic chamber on the KSW longitudinal impedance
Real part of the longitudinal impedance

38. Effect of the ceramic chamber on the KSW longitudinal impedance
Real part of the longitudinal impedance
Titanium coating
Minimum R defined by the required kicker rise time

39. Effect of the ceramic chamber on the KSW longitudinal impedance
Real part of the longitudinal impedance
Increasing the coating resistance the impedance should converge to the case without coating

40. Effect of the ceramic chamber on the KSW transverse impedance

41. Effect of the ceramic chamber on the KSW transverse impedance
Increase of the impedance due to the smaller aperture

42. Effect of the ceramic chamber on the KSW transverse impedance
The coating shifts the impedance spectrum to lower frequencies

43. Effect of the ceramic chamber on the KSW transverse impedance
The frequency shift is proportional to the coating resistance

44. Effect of the ceramic chamber on the KSW transverse impedance
Real part of the transverse impedance fr
The frequency shift is proportional to the coating resistance

45. Effect of the ceramic chamber on the KSW transverse impedance

46. KSW: transverse impedance optimization
Shift the KSW impedance spectrum out of the PSB frequency range of interest
Frequency range of interest
First possible unstable betatron line
Maximum frequency excited by the beam

47. KSW: transverse impedance optimization
Shift the KSW impedance spectrum out of the PSB frequency range of interest
Frequency range of interest
First possible unstable betatron line
Maximum frequency excited by the beam
Worst case at extraction (smallest bunch length)

48. Determination of fbeam for a generic element of impedance Z
Normalized beam power spectrum

49. Determination of fbeam: example for the KSW transverse impedance

50. KSW: transverse impedance optimization
Frequency range of interest
Shift of the KSW impedance spectrum significantly out of the PSB frequency range of interest

51. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

52. Present PSB transverse impedance model
Elements included in the database:
Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account
Extraction kicker
impedance due to the ferrite loaded structure
impedance due to the coupling to the external circuits (analytical calculation)
Indirect space charge impedance (analytical calculation)
Broadband impedance of step transitions
KSW magnets
Beam pipe
Extraction Kicker
Specificare che la cavita’ e’ quella a 200 MHz
Indirect space charge impedance
Step transitions
KSW

53. Total horizontal driving impedance of the PSB
Contributions of the extraction kicker due to the coupling with external circuits

54. Total vertical driving impedance of the PSB

55. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

56. Measurement data
D. Quatraro, Collective effects for the LHC injectors: non-ultrarelativistic approaches. PhD thesis, Bologna, University of Bologna, CERN-THESIS

57. Effective impedance of the PSB
 Zeffx,y[MΩ/m]
Ekin=160 MeV
Ekin=1.0 GeV
Ekin=1.4 GeV
Indirect space charge
1.50/10.00
0.29/1.91
0.19/1.25
Kicker cables
0.0084/0
0.012/0
0.0105/0
Kicker ferrite
-0.044/0.13
-0.04/0.11
Steps
0.53/0.63
Resistive wall
0.03/0.05
0.04/0.07
KSW
0/0.013
0/0.0013
0/0.0004
Total (expected)
2.03/10.9
0.83/2.7
0.73/2.0
Total (measured )
?/13.0
?/4.6
?/3.8
Measurements at different energies are consistent with a missing ~2 MΩ/m

58. Comparison between measurements and model of the vertical coherent tune shifts at different energies

59. Comparison between measurements and model of the vertical coherent tune shift at Ekin=160 MeV
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60. Comparison between measurements and model of the vertical coherent tune shift at Ekin=160 MeV
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61. Comparison between measurements and model of the vertical coherent tune shift at Ekin=1.0 GeV
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62. Comparison between measurements and model of the vertical coherent tune shift at Ekin=1.0 GeV
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63. Comparison between measurements and model of the vertical coherent tune shift at Ekin=1.4 GeV
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64. Comparison between measurements and model of the vertical coherent tune shift at Ekin=1.4 GeV
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30 %
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65. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

66. Summary and future plans
Measurements at different energies are consistent with a missing ~2 MΩ/m of the PSB impedance model
Measurements of the coherent horizontal and vertical tune shift
Update of the model according to new understandings and identification of significant impedance sources

67. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

68. Impedance of pumping ports

69. Impedance of pumping ports

70. Impedance of pumping ports

71. Longitudinal impedance
well above the PSB beam spectrum

72. Longitudinal impedance
well above the PSB beam spectrum

73. Transverse impedance
well above the PSB beam spectrum

74. Transverse impedance
The effective vertical impedance for 1 pumping port is expected to be well below 0.01% of the total PSB broadband impedance

75. Summary: impedance considerations on the pumping ports of the new PSB H- injection region
Obviously, a proper shielding can only be beneficial in terms of impedance, although no special issues from the beam coupling impedance point of view are expected even without shielding.

76. Overview Introduction Present PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
Resistive wall
Extraction kicker
Broadband impedance of step transitions
KSW
Global PSB impedance model
Comparison with tune shift measurements
Summary and future plans
Other devices studied
Shielding of pumping ports
Insulated flanges

77. Insulated flanges
In circular accelerators with high acceleration rate the fast variation of the main magnetic field induce currents in the ground loop.
To overcome the problem one has to cut the vacuum chamber in several sectors and reconnect them with isolated flanges.
The isolated flange forms a capacitor which inserted in series with the ground loop constitutes a parallel RLC equivalent circuit.
To shift the resonant frequency to a much lower value and to reduce the longitudinal impedance, the so called RF-bypass are connected in parallel to the flange.
R. Cappi, RF bypass on the proton synchrotron vacuum chamber flanges
Without RF-bypass
With RF-bypass

78. Beam coupling impedance of insulated flanges: example
C=1 nF
L=10 µH
R=100 Ω
Cbypass=0.4 μF
Rbypass=1 Ω

79. Insulated flanges C=0.01-100 nF L=0.5-15 µH R=1-100 Ω
Parameter sweep analysis
C= nF L= µH R=1-100 Ω
Cbypass=0.4 μF Rbypass=1 Ω
Case of study two bypasses in parallel

80. Conclusion on the beam coupling impedance of insulated flanges
The RF bypass is expected to shift the impedance spectrum to low frequency (well below the first possible unstable betatron line of the PSB) avoiding possible detrimental effect which could not be excluded in the case without RF bypass.

81. Additional impedance sources not discussed here
Vacuum chamber of the new H- injection region
Comparison between Inconel undulated chamber and titanium coated ceramic chamber
Finemet cavities
The longitudinal impedance does not depend on the relativistic beta
Foil section of the new H- injection region
Preliminary simulations do not show narrow impedances below 100 MHz

82. Thank you for your attention

83. Resistive wall impedance: impact of the iron
Resistive wall vertical generalized impedance of the PSB

84. Resistive wall impedance
A. Asner et al., The PSB main bending magnets and quadrupole lenses, Geneva, April 1970.
frel = 10 kHz
K. G. Nilanga et al., Determination of complex permeability of silicon-steel for use in high frequency modelling of power transformers, IEEE TRANS. ON MAGNETICS, VOL. 44, NO. 4, APRIL 2008.

85. PSB extraction kicker: impedance due to the coupling to the external circuits ZTEM
Horizontal
Cables in the open-short configuration

86. C-magnet: driving horizontal impedance
The C-Magnet model can support a TEM propagation
Expectation
The TEM mode plays a role when the penetration depth in the ferrite becomes comparable to the magnetic circuit length (below few hundred MHz).
Theoretical estimation
CST Particle Studio simulations
Invertire animazioni
Peak due to the TEM propagation

87. A theoretical calculation for the C-Magnet model
is the impedance calculated using the Tsutsui formalism
Constant horizontal impedance

88. C-Magnet: Comparing theoretical model and 3-D simulations
The theoretical predictions and simulations show a very good agreement
CST Particle Studio is found to be a reliable tool to simulate the impedance of ferrite loaded components
Confirmation of the new theory

89. Indirect space charge
Analytical calculation based on the PSB aperture model (provided by O. Berrig)
Rectangular pipe
Circular pipe
K. Y. Ng, Space charge impedances of beams with non-uniform transverse distributions
Elliptic pipe

90. Definition of impedance
Longitudinal component of the electric field in (x, y) induced by a source charge placed in (x0, y0)
Depend only on the source
EM simulator uses the total fields
contribution due to the interaction of beam and accelerator components

91. Overview Introduction Updated PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
KSW kickers
Longitudinal impedance model
Wake model
Other devices studied
Vacuum chamber of the new H- region
Finemet cavities
Shielding of pumping ports
Insulated flanges
Foil section
Future plans

92. Inconel undulated chamber
Alternative solution
Titanium coated Ceramic (Al2O3) chamber (no corrugation)
Vertical full aperture: 63 mm
Titanium thickness: 100 μm
Inconel thickness: mm
Vertical full aperture: 63 mm
Inconel conductivity =

93. Analytical calculation (no corrugation): comparison between Inconel and Ceramic chamber
Theoretical calculation made with the TLwall code

94. Inconel undulated chamber: CST Particle Studio simulation
The impedance contribution of the corrugation seems to be negligible

95. Overview Introduction Updated PSB impedance model
Indirect space charge
Numerical form factor for ISC computation
KSW kickers
Longitudinal impedance model
Wake model
Other devices studied
Vacuum chamber of the new H- region
Finemet cavities
Shielding of pumping ports
Insulated flanges
Foil section
Future plans

96. Finemet cavities Ps and PSB cell cavity
The impedance does not depend on the relativistic beta
S. Persichelli et al., Finemet cavities impedance studies, CERN-ACC-NOTE , 2013.